LMFDB - Newform orbit 4704.2.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(2353,4704)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4704.2353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 $ x^16 + x^14 - 2x^13 - 2x^12 - 4x^11 - 2x^10 + 16x^9 + 8x^8 + 32x^7 - 8x^6 - 32x^5 - 32x^4 - 64x^3 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{9}+O(q^{10}) $ q + b2 * q^3 + b1 * q^5 - q^9	$ q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{9} + \beta_{13} q^{11} + \beta_{12} q^{13} - \beta_{3} q^{15} + ( - \beta_{5} - \beta_{3}) q^{17} + (\beta_{13} - \beta_{7} - \beta_{4} - \beta_1) q^{19} + (\beta_{11} - 2 \beta_{6} + \beta_{5} + \cdots - 1) q^{23}+ \cdots - \beta_{13} q^{99}+O(q^{100}) $ q + b2 * q^3 + b1 * q^5 - q^9 + b13 * q^11 + b12 * q^13 - b3 * q^15 + (-b5 - b3) * q^17 + (b13 - b7 - b4 - b1) * q^19 + (b11 - 2b6 + b5 - b3 - 1) q^23 + (b10 - b9 + b6 + b3 - 1) * q^25 - b2 * q^27 + (-b14 - b4 + b2) * q^29 + (b11 + b9 + b8 - b5 - b3 + 1) * q^31 + b6 * q^33 + (-b15 - 2b13 - 2b2 + b1) * q^37 + b9 * q^39 + (b10 + b9 + b3) * q^41 + (b15 + b13 - b12 + b7 - b4) * q^43 - b1 * q^45 + (b11 + b10 - b9 + b5 + 1) * q^47 + (b14 - b1) * q^51 + (-b15 - b14 + b12 - b7 - b4 - 3b2) q^53 + (-b8 + 2b6 - 2) q^55 + (-b11 - b8 + b6 + b3) * q^57 + (2b15 + b13 + b7 + b4 + b2 - b1) q^59 + (b15 + 2b14 - b12 - b1) q^61 + (-b10 - b9 - 2b8 + b5 + 2b3) * q^65 + (2b15 + 2b14 - b13 + b7 + b4 + 2b2 + b1) q^67 + (-b14 + 2b13 - b4 - b2 - b1) q^69 + (-b11 - b10 + b9 - 2b8 + b5 + 3) q^71 + (b11 + b8 + b6 - b5 + 2b3) q^73 + (-b15 - b13 + b12 - b2 + b1) * q^75 + (-b11 + b9 - b8 + 2b6 + b5 + b3 + 1) q^79 + q^81 + (b15 + b14 - b13 - b12 + 2b7) q^83 + (-b15 + b12 - 2b4 - 6b2 - b1) * q^85 + (-b11 - b5 - 1) * q^87 + (-2b10 + 2b9 - 2b8) q^89 + (b14 - b12 - b7 - b4 + b2 - b1) * q^93 + (b11 - b10 + b9 + b5 - 2b3 + 1) q^95 + (b11 - 2b9 - b8 - b6 + b5 - 2b3 + 1) * q^97 - b13 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 8 q^{23} - 16 q^{25} + 24 q^{31} + 24 q^{47} - 32 q^{55} - 8 q^{57} + 40 q^{71} + 8 q^{73} + 8 q^{79} + 16 q^{81} - 24 q^{87} + 24 q^{95} + 24 q^{97}+O(q^{100}) $ 16 * q - 16 * q^9 - 8 * q^23 - 16 * q^25 + 24 * q^31 + 24 * q^47 - 32 * q^55 - 8 * q^57 + 40 * q^71 + 8 * q^73 + 8 * q^79 + 16 * q^81 - 24 * q^87 + 24 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 $ x^16 + x^14 - 2*x^13 - 2*x^12 - 4*x^11 - 2*x^10 + 16*x^9 + 8*x^8 + 32*x^7 - 8*x^6 - 32*x^5 - 32*x^4 - 64*x^3 + 64*x^2 + 256 :

$\beta_{1}$	$=$	$ ( \nu^{15} - 2 \nu^{14} + \nu^{13} - 4 \nu^{12} - 6 \nu^{11} + 16 \nu^{10} + 14 \nu^{9} + 20 \nu^{8} + \cdots + 384 ) / 256 $ (v^15 - 2v^14 + v^13 - 4v^12 - 6v^11 + 16v^10 + 14v^9 + 20v^8 + 8v^7 + 16v^6 - 120v^5 - 176v^4 + 64v^3 - 64v^2 + 320*v + 384) / 256
$\beta_{2}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - \nu^{13} - 4 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} + \cdots - 384 ) / 256 $ (-v^15 + 2v^14 - v^13 - 4v^12 + 6v^11 - 8v^10 + 2v^9 - 20v^8 - 8v^7 - 8v^5 + 80v^4 - 64v^3 + 64v^2 - 64v - 384) / 256
$\beta_{3}$	$=$	$ ( - \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 2 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + \cdots - 64 ) / 64 $ (-v^14 + 2v^13 - 5v^12 + 2v^10 - 4v^9 + 18v^8 - 4v^7 + 16v^6 - 40v^5 - 8v^4 - 128v^2 + 192*v - 64) / 64
$\beta_{4}$	$=$	$ ( - 3 \nu^{15} - 2 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + 2 \nu^{11} + 24 \nu^{10} + 6 \nu^{9} + \cdots + 384 ) / 256 $ (-3v^15 - 2v^14 - 3v^13 + 12v^12 + 2v^11 + 24v^10 + 6v^9 - 44v^8 - 24v^7 - 128v^6 + 40v^5 + 48v^4 + 192v^3 + 192v^2 - 704*v + 384) / 256
$\beta_{5}$	$=$	$ ( - \nu^{15} + 4 \nu^{14} - 9 \nu^{13} + 6 \nu^{12} - 6 \nu^{11} - 4 \nu^{10} + 26 \nu^{9} - 24 \nu^{8} + \cdots + 512 ) / 128 $ (-v^15 + 4v^14 - 9v^13 + 6v^12 - 6v^11 - 4v^10 + 26v^9 - 24v^8 + 72v^7 - 96v^6 - 8v^5 - 32v^4 - 128v^3 + 448v^2 - 192v + 512) / 128
$\beta_{6}$	$=$	$ ( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 10 \nu^{11} - 12 \nu^{10} + 26 \nu^{9} + \cdots + 512 ) / 128 $ (-v^15 + 4v^14 - 5v^13 + 6v^12 - 10v^11 - 12v^10 + 26v^9 - 24v^8 + 64v^7 - 32v^6 + 40v^5 - 32v^4 - 256v^3 + 320v^2 - 320v + 512) / 128
$\beta_{7}$	$=$	$ ( 5 \nu^{15} - 2 \nu^{14} - 3 \nu^{13} - 12 \nu^{12} - 6 \nu^{11} + 16 \nu^{10} + 6 \nu^{9} + 84 \nu^{8} + \cdots - 640 ) / 256 $ (5v^15 - 2v^14 - 3v^13 - 12v^12 - 6v^11 + 16v^10 + 6v^9 + 84v^8 - 72v^7 + 16v^6 - 120v^5 - 48v^4 + 192v^3 - 64v^2 + 576*v - 640) / 256
$\beta_{8}$	$=$	$ ( \nu^{15} + 4 \nu^{14} - 7 \nu^{13} + 2 \nu^{12} - 10 \nu^{11} - 12 \nu^{10} + 22 \nu^{9} + 8 \nu^{8} + \cdots + 512 ) / 128 $ (v^15 + 4v^14 - 7v^13 + 2v^12 - 10v^11 - 12v^10 + 22v^9 + 8v^8 + 88v^7 - 32v^6 - 24v^5 - 96v^4 - 192v^3 + 320v^2 - 320v + 512) / 128
$\beta_{9}$	$=$	$ ( 3 \nu^{14} - 4 \nu^{13} + 3 \nu^{12} - 10 \nu^{11} - 6 \nu^{10} + 12 \nu^{9} + 2 \nu^{8} + 56 \nu^{7} + \cdots + 384 ) / 64 $ (3v^14 - 4v^13 + 3v^12 - 10v^11 - 6v^10 + 12v^9 + 2v^8 + 56v^7 - 24v^6 + 32v^5 - 104v^4 - 96v^3 + 96v^2 - 128v + 384) / 64
$\beta_{10}$	$=$	$ ( - \nu^{15} - 6 \nu^{14} - \nu^{13} - 4 \nu^{12} + 6 \nu^{11} + 16 \nu^{10} + 2 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 128 $ (-v^15 - 6v^14 - v^13 - 4v^12 + 6v^11 + 16v^10 + 2v^9 + 12v^8 - 72v^7 - 16v^6 - 40v^5 - 48v^4 + 192v^3 - 128v^2 + 192*v - 256) / 128
$\beta_{11}$	$=$	$ ( \nu^{15} - 2 \nu^{14} + 13 \nu^{13} - 12 \nu^{12} - 2 \nu^{11} - 18 \nu^{9} + 52 \nu^{8} - 48 \nu^{7} + \cdots - 256 ) / 128 $ (v^15 - 2v^14 + 13v^13 - 12v^12 - 2v^11 - 18v^9 + 52v^8 - 48v^7 + 96v^6 - 72v^5 - 16v^4 + 64v^3 - 320v^2 + 704*v - 256) / 128
$\beta_{12}$	$=$	$ ( 3 \nu^{15} - 4 \nu^{14} + 3 \nu^{13} - 2 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} + 2 \nu^{9} + 24 \nu^{8} + \cdots + 128 ) / 128 $ (3v^15 - 4v^14 + 3v^13 - 2v^12 - 6v^11 + 4v^10 + 2v^9 + 24v^8 - 8v^7 + 16v^6 - 8v^5 - 64v^4 + 192v^3 - 192v^2 + 64*v + 128) / 128
$\beta_{13}$	$=$	$ ( - 7 \nu^{15} + 2 \nu^{14} + \nu^{13} + 8 \nu^{12} + 26 \nu^{11} - 2 \nu^{9} - 84 \nu^{8} - 72 \nu^{7} + \cdots - 640 ) / 256 $ (-7v^15 + 2v^14 + v^13 + 8v^12 + 26v^11 - 2v^9 - 84v^8 - 72v^7 - 24v^5 + 400v^4 + 64v^3 + 64v^2 - 192v - 640) / 256
$\beta_{14}$	$=$	$ ( - 5 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} - 4 \nu^{12} + 46 \nu^{11} + 16 \nu^{10} + 58 \nu^{9} + \cdots - 896 ) / 256 $ (-5v^15 + 2v^14 - 5v^13 - 4v^12 + 46v^11 + 16v^10 + 58v^9 - 20v^8 - 104v^7 - 112v^6 - 168v^5 + 368v^4 + 64v^3 + 576v^2 + 704*v - 896) / 256
$\beta_{15}$	$=$	$ ( 7 \nu^{15} - 2 \nu^{14} + 7 \nu^{13} - 34 \nu^{11} + 8 \nu^{10} - 14 \nu^{9} + 52 \nu^{8} + \cdots + 1152 ) / 256 $ (7v^15 - 2v^14 + 7v^13 - 34v^11 + 8v^10 - 14v^9 + 52v^8 + 120v^7 + 112v^6 + 120v^5 - 368v^4 + 192v^3 - 192v^2 - 64v + 1152) / 256

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} ) / 4 $ (-b8 + b5 - b4 - b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{3} - \beta _1 - 1 ) / 4 $ (b15 - b12 + b11 + b10 - b9 + b8 + b5 - b3 - b1 - 1) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + 2 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{6} + \cdots + 1 ) / 4 $ (b15 + 2b13 + b12 + b11 + b10 + b9 + b8 - 2b6 + b5 - b3 + 2*b2 - b1 + 1) / 4
$\nu^{4}$	$=$	$ ( \beta_{15} + 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 3 ) / 4 $ (b15 + 2b13 + b12 - b11 - b10 - b9 - b8 + 2b7 + 2b6 + b5 + 3b3 - 2b2 - 3b1 + 3) / 4
$\nu^{5}$	$=$	$ ( 3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} + \cdots + 3 ) / 4 $ (3b15 + 2b14 - 2b13 - b12 - b11 + b10 + b9 - 3b8 + 2b6 - b5 + b3 + 2b2 - 5*b1 + 3) / 4
$\nu^{6}$	$=$	$ ( 5 \beta_{15} + 6 \beta_{13} - 3 \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} + \cdots + 3 ) / 4 $ (5b15 + 6b13 - 3b12 - b11 + 3b10 - b9 + 3b8 + 2b7 + 2b6 - 3b5 - 4b4 - b3 - 2b2 + b1 + 3) / 4
$\nu^{7}$	$=$	$ ( 5 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 23 ) / 4 $ (5b15 + 2b14 + 2b13 + b12 - 3b11 - b10 - b9 + b8 - 4b7 - 2b6 + 3b5 - 2b4 + 7b3 - 4b2 - 3*b1 - 23) / 4
$\nu^{8}$	$=$	$ ( \beta_{15} + 4 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + \cdots - 5 ) / 4 $ (b15 + 4b14 + 2b13 - 3b12 + 3b11 + 3b10 + 3b9 + 11b8 + 2b7 - 2b6 - 7b5 + 4b4 + 3b3 - 22b2 - 15b1 - 5) / 4
$\nu^{9}$	$=$	$ ( \beta_{15} + 10 \beta_{14} + 2 \beta_{13} + 13 \beta_{12} - 3 \beta_{11} - \beta_{10} + 7 \beta_{9} + \cdots - 15 ) / 4 $ (b15 + 10b14 + 2b13 + 13b12 - 3b11 - b10 + 7b9 - 3b8 + 14b6 - 13b5 + 2b4 + 3b3 + 8b2 + 5b1 - 15) / 4
$\nu^{10}$	$=$	$ ( 17 \beta_{15} + 8 \beta_{14} - 6 \beta_{13} - 19 \beta_{12} - 17 \beta_{11} - 13 \beta_{10} - 5 \beta_{9} + \cdots - 17 ) / 4 $ (17b15 + 8b14 - 6b13 - 19b12 - 17b11 - 13b10 - 5b9 - 17b8 + 14b7 + 14b6 - 11b5 + 16b4 + 19b3 - 2b2 + 17*b1 - 17) / 4
$\nu^{11}$	$=$	$ ( - 3 \beta_{15} + 22 \beta_{14} - 22 \beta_{13} - 7 \beta_{12} - 11 \beta_{11} - 5 \beta_{10} + \cdots + 33 ) / 4 $ (-3b15 + 22b14 - 22b13 - 7b12 - 11b11 - 5b10 + 3b9 + 17b8 - 28b7 - 26b6 - 25b5 - 6b4 - 9b3 - 8b2 + 9*b1 + 33) / 4
$\nu^{12}$	$=$	$ ( - 31 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 13 \beta_{10} + \cdots - 125 ) / 4 $ (-31b15 - 4b14 + 18b13 + 5b12 + 3b11 - 13b10 + 3b9 + 11b8 - 6b7 - 10b6 - 7b5 - 20b4 - 29b3 - 158b2 + 17*b1 - 125) / 4
$\nu^{13}$	$=$	$ ( - 11 \beta_{15} + 18 \beta_{14} - 38 \beta_{13} + 17 \beta_{12} + 41 \beta_{11} - 29 \beta_{10} + \cdots - 131 ) / 4 $ (-11b15 + 18b14 - 38b13 + 17b12 + 41b11 - 29b10 - 5b9 + 5b8 - 40b7 + 6b6 - 13b5 + 54b4 - 17b3 + 20b2 + 17*b1 - 131) / 4
$\nu^{14}$	$=$	$ ( - 15 \beta_{15} + 16 \beta_{14} + 2 \beta_{13} - 27 \beta_{12} + 15 \beta_{11} - 69 \beta_{10} + \cdots + 167 ) / 4 $ (-15b15 + 16b14 + 2b13 - 27b12 + 15b11 - 69b10 + 75b9 - 17b8 + 46b7 - 58b6 - 51b5 + 56b4 - 45b3 + 126b2 + 73*b1 + 167) / 4
$\nu^{15}$	$=$	$ ( - 51 \beta_{15} + 6 \beta_{14} - 118 \beta_{13} + 57 \beta_{12} - 67 \beta_{11} - 125 \beta_{10} + \cdots + 105 ) / 4 $ (-51b15 + 6b14 - 118b13 + 57b12 - 67b11 - 125b10 - 53b9 - 71b8 + 28b7 + 6b6 - 9b5 - 22b4 - 41b3 + 56b2 + 113*b1 + 105) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$1$	$-1$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

2353.1

−0.236856	+	1.39424i
0.284419	−	1.38532i
−0.925802	−	1.06906i
1.36937	+	0.353295i
−1.39783	−	0.214614i
0.462163	+	1.33656i
1.30955	−	0.533922i
−0.865014	+	1.11882i
−0.865014	−	1.11882i
1.30955	+	0.533922i
0.462163	−	1.33656i
−1.39783	+	0.214614i
1.36937	−	0.353295i
−0.925802	+	1.06906i
0.284419	+	1.38532i
−0.236856	−	1.39424i

−

1.00000i

−

3.56550i

−1.00000

2353.2

−

1.00000i

−

2.29465i

−1.00000

2353.3

−

1.00000i

−

1.80422i

−1.00000

2353.4

−

1.00000i

−

0.677172i

−1.00000

2353.5

−

1.00000i

−

0.0464447i

−1.00000

2353.6

−

1.00000i

1.42114i

−1.00000

2353.7

−

1.00000i

3.38908i

−1.00000

2353.8

−

1.00000i

3.57776i

−1.00000

2353.9

1.00000i

−

3.57776i

−1.00000

2353.10

1.00000i

−

3.38908i

−1.00000

2353.11

1.00000i

−

1.42114i

−1.00000

2353.12

1.00000i

0.0464447i

−1.00000

2353.13

1.00000i

0.677172i

−1.00000

2353.14

1.00000i

1.80422i

−1.00000

2353.15

1.00000i

2.29465i

−1.00000

2353.16

1.00000i

3.56550i

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4704.2.c.f		16
4.b	odd	2	1		1176.2.c.f		16
7.b	odd	2	1		4704.2.c.e		16
7.d	odd	6	2		672.2.bk.a		32
8.b	even	2	1	inner	4704.2.c.f		16
8.d	odd	2	1		1176.2.c.f		16
21.g	even	6	2		2016.2.cr.e		32
28.d	even	2	1		1176.2.c.e		16
28.f	even	6	2		168.2.bc.a	✓	32
56.e	even	2	1		1176.2.c.e		16
56.h	odd	2	1		4704.2.c.e		16
56.j	odd	6	2		672.2.bk.a		32
56.m	even	6	2		168.2.bc.a	✓	32
84.j	odd	6	2		504.2.cj.e		32
168.ba	even	6	2		2016.2.cr.e		32
168.be	odd	6	2		504.2.cj.e		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.bc.a	✓	32	28.f	even	6	2
168.2.bc.a	✓	32	56.m	even	6	2
504.2.cj.e		32	84.j	odd	6	2
504.2.cj.e		32	168.be	odd	6	2
672.2.bk.a		32	7.d	odd	6	2
672.2.bk.a		32	56.j	odd	6	2
1176.2.c.e		16	28.d	even	2	1
1176.2.c.e		16	56.e	even	2	1
1176.2.c.f		16	4.b	odd	2	1
1176.2.c.f		16	8.d	odd	2	1
2016.2.cr.e		32	21.g	even	6	2
2016.2.cr.e		32	168.ba	even	6	2
4704.2.c.e		16	7.b	odd	2	1
4704.2.c.e		16	56.h	odd	2	1
4704.2.c.f		16	1.a	even	1	1	trivial
4704.2.c.f		16	8.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4704, [\chi])$:

$ T_{5}^{16} + 48T_{5}^{14} + 902T_{5}^{12} + 8384T_{5}^{10} + 40313T_{5}^{8} + 96864T_{5}^{6} + 101584T_{5}^{4} + 29888T_{5}^{2} + 64 $

$ T_{17}^{8} - 76T_{17}^{6} + 104T_{17}^{5} + 1544T_{17}^{4} - 3744T_{17}^{3} - 1184T_{17}^{2} + 896T_{17} + 256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$5$	$ T^{16} + 48 T^{14} + \cdots + 64 $ T^16 + 48T^14 + 902T^12 + 8384T^10 + 40313T^8 + 96864T^6 + 101584T^4 + 29888*T^2 + 64
$7$	$ T^{16} $ T^16
$11$	$ T^{16} + 76 T^{14} + \cdots + 2304 $ T^16 + 76T^14 + 2190T^12 + 29900T^10 + 194425T^8 + 528720T^6 + 563808T^4 + 205056*T^2 + 2304
$13$	$ T^{16} + 108 T^{14} + \cdots + 1149184 $ T^16 + 108T^14 + 4182T^12 + 71692T^10 + 582033T^8 + 2221888T^6 + 4161888T^4 + 3646464*T^2 + 1149184
$17$	$ (T^{8} - 76 T^{6} + \cdots + 256)^{2} $ (T^8 - 76T^6 + 104T^5 + 1544T^4 - 3744T^3 - 1184T^2 + 896T + 256)^2
$19$	$ T^{16} + 140 T^{14} + \cdots + 11723776 $ T^16 + 140T^14 + 7478T^12 + 197300T^10 + 2784513T^8 + 21381384T^6 + 84436048T^4 + 136476160*T^2 + 11723776
$23$	$ (T^{8} + 4 T^{7} + \cdots + 25472)^{2} $ (T^8 + 4T^7 - 120T^6 - 568T^5 + 3544T^4 + 18176T^3 - 17792T^2 - 96384*T + 25472)^2
$29$	$ T^{16} + 192 T^{14} + \cdots + 589824 $ T^16 + 192T^14 + 14086T^12 + 492304T^10 + 8249353T^8 + 58264896T^6 + 145390080T^4 + 29048832*T^2 + 589824
$31$	$ (T^{8} - 12 T^{7} + \cdots - 145241)^{2} $ (T^8 - 12T^7 - 90T^6 + 1168T^5 + 2448T^4 - 32204T^3 - 24462T^2 + 222264*T - 145241)^2
$37$	$ T^{16} + \cdots + 13545235456 $ T^16 + 324T^14 + 39174T^12 + 2275324T^10 + 67664817T^8 + 995045944T^6 + 6339030480T^4 + 16189502976*T^2 + 13545235456
$41$	$ (T^{8} - 148 T^{6} + \cdots - 29696)^{2} $ (T^8 - 148T^6 + 352T^5 + 5936T^4 - 25344T^3 - 12992T^2 + 128512T - 29696)^2
$43$	$ T^{16} + \cdots + 326726560000 $ T^16 + 372T^14 + 54566T^12 + 4099172T^10 + 170784449T^8 + 3965288880T^6 + 48310589920T^4 + 258970229504*T^2 + 326726560000
$47$	$ (T^{8} - 12 T^{7} + \cdots - 2304)^{2} $ (T^8 - 12T^7 - 60T^6 + 1048T^5 - 568T^4 - 23328T^3 + 54816T^2 + 21888*T - 2304)^2
$53$	$ T^{16} + \cdots + 1973491776 $ T^16 + 304T^14 + 37174T^12 + 2358576T^10 + 82621705T^8 + 1541139312T^6 + 12750847632T^4 + 19458467136*T^2 + 1973491776
$59$	$ T^{16} + \cdots + 3515066944 $ T^16 + 476T^14 + 90910T^12 + 8856236T^10 + 459972409T^8 + 11969866592T^6 + 125121353104T^4 + 262062897728*T^2 + 3515066944
$61$	$ T^{16} + \cdots + 10909384704 $ T^16 + 408T^14 + 52048T^12 + 2937856T^10 + 79768576T^8 + 1023442944T^6 + 5719130112T^4 + 13627293696*T^2 + 10909384704
$67$	$ T^{16} + \cdots + 499498389504 $ T^16 + 636T^14 + 148534T^12 + 16011508T^10 + 830877313T^8 + 20407019544T^6 + 231892894224T^4 + 1017100924416*T^2 + 499498389504
$71$	$ (T^{8} - 20 T^{7} + \cdots + 8590080)^{2} $ (T^8 - 20T^7 - 60T^6 + 3912T^5 - 23512T^4 - 107424T^3 + 1681824T^2 - 6526848*T + 8590080)^2
$73$	$ (T^{8} - 4 T^{7} + \cdots - 1022064)^{2} $ (T^8 - 4T^7 - 290T^6 + 340T^5 + 27137T^4 + 33048T^3 - 756072T^2 - 2231136*T - 1022064)^2
$79$	$ (T^{8} - 4 T^{7} + \cdots - 37585)^{2} $ (T^8 - 4T^7 - 290T^6 + 2080T^5 + 8104T^4 - 63876T^3 - 5670T^2 + 140984*T - 37585)^2
$83$	$ T^{16} + \cdots + 29873665600 $ T^16 + 588T^14 + 114462T^12 + 10092844T^10 + 425922873T^8 + 7641426736T^6 + 33130946256T^4 + 53846383296*T^2 + 29873665600
$89$	$ (T^{8} - 336 T^{6} + \cdots + 1114112)^{2} $ (T^8 - 336T^6 - 576T^5 + 21120T^4 + 34816T^3 - 354304T^2 - 196608T + 1114112)^2
$97$	$ (T^{8} - 12 T^{7} + \cdots - 2509616)^{2} $ (T^8 - 12T^7 - 338T^6 + 2956T^5 + 31313T^4 - 237448T^3 - 729192T^2 + 5650592*T - 2509616)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{9}+O(q^{10}) \) q + b2 * q^3 + b1 * q^5 - q^9	\( q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{9} + \beta_{13} q^{11} + \beta_{12} q^{13} - \beta_{3} q^{15} + ( - \beta_{5} - \beta_{3}) q^{17} + (\beta_{13} - \beta_{7} - \beta_{4} - \beta_1) q^{19} + (\beta_{11} - 2 \beta_{6} + \beta_{5} + \cdots - 1) q^{23}+ \cdots - \beta_{13} q^{99}+O(q^{100}) \) q + b2 * q^3 + b1 * q^5 - q^9 + b13 * q^11 + b12 * q^13 - b3 * q^15 + (-b5 - b3) * q^17 + (b13 - b7 - b4 - b1) * q^19 + (b11 - 2b6 + b5 - b3 - 1) q^23 + (b10 - b9 + b6 + b3 - 1) * q^25 - b2 * q^27 + (-b14 - b4 + b2) * q^29 + (b11 + b9 + b8 - b5 - b3 + 1) * q^31 + b6 * q^33 + (-b15 - 2b13 - 2b2 + b1) * q^37 + b9 * q^39 + (b10 + b9 + b3) * q^41 + (b15 + b13 - b12 + b7 - b4) * q^43 - b1 * q^45 + (b11 + b10 - b9 + b5 + 1) * q^47 + (b14 - b1) * q^51 + (-b15 - b14 + b12 - b7 - b4 - 3b2) q^53 + (-b8 + 2b6 - 2) q^55 + (-b11 - b8 + b6 + b3) * q^57 + (2b15 + b13 + b7 + b4 + b2 - b1) q^59 + (b15 + 2b14 - b12 - b1) q^61 + (-b10 - b9 - 2b8 + b5 + 2b3) * q^65 + (2b15 + 2b14 - b13 + b7 + b4 + 2b2 + b1) q^67 + (-b14 + 2b13 - b4 - b2 - b1) q^69 + (-b11 - b10 + b9 - 2b8 + b5 + 3) q^71 + (b11 + b8 + b6 - b5 + 2b3) q^73 + (-b15 - b13 + b12 - b2 + b1) * q^75 + (-b11 + b9 - b8 + 2b6 + b5 + b3 + 1) q^79 + q^81 + (b15 + b14 - b13 - b12 + 2b7) q^83 + (-b15 + b12 - 2b4 - 6b2 - b1) * q^85 + (-b11 - b5 - 1) * q^87 + (-2b10 + 2b9 - 2b8) q^89 + (b14 - b12 - b7 - b4 + b2 - b1) * q^93 + (b11 - b10 + b9 + b5 - 2b3 + 1) q^95 + (b11 - 2b9 - b8 - b6 + b5 - 2b3 + 1) * q^97 - b13 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 8 q^{23} - 16 q^{25} + 24 q^{31} + 24 q^{47} - 32 q^{55} - 8 q^{57} + 40 q^{71} + 8 q^{73} + 8 q^{79} + 16 q^{81} - 24 q^{87} + 24 q^{95} + 24 q^{97}+O(q^{100}) \) 16 * q - 16 * q^9 - 8 * q^23 - 16 * q^25 + 24 * q^31 + 24 * q^47 - 32 * q^55 - 8 * q^57 + 40 * q^71 + 8 * q^73 + 8 * q^79 + 16 * q^81 - 24 * q^87 + 24 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4704.2.c.f

Level	$4704$
Weight	$2$
Character orbit	4704.c
Analytic conductor	$37.562$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(37.5616291108\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 \) x^16 + x^14 - 2x^13 - 2x^12 - 4x^11 - 2x^10 + 16x^9 + 8x^8 + 32x^7 - 8x^6 - 32x^5 - 32x^4 - 64x^3 + 64x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{13} \)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} - 2 \nu^{14} + \nu^{13} - 4 \nu^{12} - 6 \nu^{11} + 16 \nu^{10} + 14 \nu^{9} + 20 \nu^{8} + \cdots + 384 ) / 256 \) (v^15 - 2v^14 + v^13 - 4v^12 - 6v^11 + 16v^10 + 14v^9 + 20v^8 + 8v^7 + 16v^6 - 120v^5 - 176v^4 + 64v^3 - 64v^2 + 320*v + 384) / 256
\(\beta_{2}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - \nu^{13} - 4 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} + \cdots - 384 ) / 256 \) (-v^15 + 2v^14 - v^13 - 4v^12 + 6v^11 - 8v^10 + 2v^9 - 20v^8 - 8v^7 - 8v^5 + 80v^4 - 64v^3 + 64v^2 - 64v - 384) / 256
\(\beta_{3}\)	\(=\)	\( ( - \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 2 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + \cdots - 64 ) / 64 \) (-v^14 + 2v^13 - 5v^12 + 2v^10 - 4v^9 + 18v^8 - 4v^7 + 16v^6 - 40v^5 - 8v^4 - 128v^2 + 192*v - 64) / 64
\(\beta_{4}\)	\(=\)	\( ( - 3 \nu^{15} - 2 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + 2 \nu^{11} + 24 \nu^{10} + 6 \nu^{9} + \cdots + 384 ) / 256 \) (-3v^15 - 2v^14 - 3v^13 + 12v^12 + 2v^11 + 24v^10 + 6v^9 - 44v^8 - 24v^7 - 128v^6 + 40v^5 + 48v^4 + 192v^3 + 192v^2 - 704*v + 384) / 256
\(\beta_{5}\)	\(=\)	\( ( - \nu^{15} + 4 \nu^{14} - 9 \nu^{13} + 6 \nu^{12} - 6 \nu^{11} - 4 \nu^{10} + 26 \nu^{9} - 24 \nu^{8} + \cdots + 512 ) / 128 \) (-v^15 + 4v^14 - 9v^13 + 6v^12 - 6v^11 - 4v^10 + 26v^9 - 24v^8 + 72v^7 - 96v^6 - 8v^5 - 32v^4 - 128v^3 + 448v^2 - 192v + 512) / 128
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 10 \nu^{11} - 12 \nu^{10} + 26 \nu^{9} + \cdots + 512 ) / 128 \) (-v^15 + 4v^14 - 5v^13 + 6v^12 - 10v^11 - 12v^10 + 26v^9 - 24v^8 + 64v^7 - 32v^6 + 40v^5 - 32v^4 - 256v^3 + 320v^2 - 320v + 512) / 128
\(\beta_{7}\)	\(=\)	\( ( 5 \nu^{15} - 2 \nu^{14} - 3 \nu^{13} - 12 \nu^{12} - 6 \nu^{11} + 16 \nu^{10} + 6 \nu^{9} + 84 \nu^{8} + \cdots - 640 ) / 256 \) (5v^15 - 2v^14 - 3v^13 - 12v^12 - 6v^11 + 16v^10 + 6v^9 + 84v^8 - 72v^7 + 16v^6 - 120v^5 - 48v^4 + 192v^3 - 64v^2 + 576*v - 640) / 256
\(\beta_{8}\)	\(=\)	\( ( \nu^{15} + 4 \nu^{14} - 7 \nu^{13} + 2 \nu^{12} - 10 \nu^{11} - 12 \nu^{10} + 22 \nu^{9} + 8 \nu^{8} + \cdots + 512 ) / 128 \) (v^15 + 4v^14 - 7v^13 + 2v^12 - 10v^11 - 12v^10 + 22v^9 + 8v^8 + 88v^7 - 32v^6 - 24v^5 - 96v^4 - 192v^3 + 320v^2 - 320v + 512) / 128
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{14} - 4 \nu^{13} + 3 \nu^{12} - 10 \nu^{11} - 6 \nu^{10} + 12 \nu^{9} + 2 \nu^{8} + 56 \nu^{7} + \cdots + 384 ) / 64 \) (3v^14 - 4v^13 + 3v^12 - 10v^11 - 6v^10 + 12v^9 + 2v^8 + 56v^7 - 24v^6 + 32v^5 - 104v^4 - 96v^3 + 96v^2 - 128v + 384) / 64
\(\beta_{10}\)	\(=\)	\( ( - \nu^{15} - 6 \nu^{14} - \nu^{13} - 4 \nu^{12} + 6 \nu^{11} + 16 \nu^{10} + 2 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 128 \) (-v^15 - 6v^14 - v^13 - 4v^12 + 6v^11 + 16v^10 + 2v^9 + 12v^8 - 72v^7 - 16v^6 - 40v^5 - 48v^4 + 192v^3 - 128v^2 + 192*v - 256) / 128
\(\beta_{11}\)	\(=\)	\( ( \nu^{15} - 2 \nu^{14} + 13 \nu^{13} - 12 \nu^{12} - 2 \nu^{11} - 18 \nu^{9} + 52 \nu^{8} - 48 \nu^{7} + \cdots - 256 ) / 128 \) (v^15 - 2v^14 + 13v^13 - 12v^12 - 2v^11 - 18v^9 + 52v^8 - 48v^7 + 96v^6 - 72v^5 - 16v^4 + 64v^3 - 320v^2 + 704*v - 256) / 128
\(\beta_{12}\)	\(=\)	\( ( 3 \nu^{15} - 4 \nu^{14} + 3 \nu^{13} - 2 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} + 2 \nu^{9} + 24 \nu^{8} + \cdots + 128 ) / 128 \) (3v^15 - 4v^14 + 3v^13 - 2v^12 - 6v^11 + 4v^10 + 2v^9 + 24v^8 - 8v^7 + 16v^6 - 8v^5 - 64v^4 + 192v^3 - 192v^2 + 64*v + 128) / 128
\(\beta_{13}\)	\(=\)	\( ( - 7 \nu^{15} + 2 \nu^{14} + \nu^{13} + 8 \nu^{12} + 26 \nu^{11} - 2 \nu^{9} - 84 \nu^{8} - 72 \nu^{7} + \cdots - 640 ) / 256 \) (-7v^15 + 2v^14 + v^13 + 8v^12 + 26v^11 - 2v^9 - 84v^8 - 72v^7 - 24v^5 + 400v^4 + 64v^3 + 64v^2 - 192v - 640) / 256
\(\beta_{14}\)	\(=\)	\( ( - 5 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} - 4 \nu^{12} + 46 \nu^{11} + 16 \nu^{10} + 58 \nu^{9} + \cdots - 896 ) / 256 \) (-5v^15 + 2v^14 - 5v^13 - 4v^12 + 46v^11 + 16v^10 + 58v^9 - 20v^8 - 104v^7 - 112v^6 - 168v^5 + 368v^4 + 64v^3 + 576v^2 + 704*v - 896) / 256
\(\beta_{15}\)	\(=\)	\( ( 7 \nu^{15} - 2 \nu^{14} + 7 \nu^{13} - 34 \nu^{11} + 8 \nu^{10} - 14 \nu^{9} + 52 \nu^{8} + \cdots + 1152 ) / 256 \) (7v^15 - 2v^14 + 7v^13 - 34v^11 + 8v^10 - 14v^9 + 52v^8 + 120v^7 + 112v^6 + 120v^5 - 368v^4 + 192v^3 - 192v^2 - 64v + 1152) / 256

\(\nu\)	\(=\)	\( ( -\beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} ) / 4 \) (-b8 + b5 - b4 - b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{3} - \beta _1 - 1 ) / 4 \) (b15 - b12 + b11 + b10 - b9 + b8 + b5 - b3 - b1 - 1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 2 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{6} + \cdots + 1 ) / 4 \) (b15 + 2b13 + b12 + b11 + b10 + b9 + b8 - 2b6 + b5 - b3 + 2*b2 - b1 + 1) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} + 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 3 ) / 4 \) (b15 + 2b13 + b12 - b11 - b10 - b9 - b8 + 2b7 + 2b6 + b5 + 3b3 - 2b2 - 3b1 + 3) / 4
\(\nu^{5}\)	\(=\)	\( ( 3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} + \cdots + 3 ) / 4 \) (3b15 + 2b14 - 2b13 - b12 - b11 + b10 + b9 - 3b8 + 2b6 - b5 + b3 + 2b2 - 5*b1 + 3) / 4
\(\nu^{6}\)	\(=\)	\( ( 5 \beta_{15} + 6 \beta_{13} - 3 \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} + \cdots + 3 ) / 4 \) (5b15 + 6b13 - 3b12 - b11 + 3b10 - b9 + 3b8 + 2b7 + 2b6 - 3b5 - 4b4 - b3 - 2b2 + b1 + 3) / 4
\(\nu^{7}\)	\(=\)	\( ( 5 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 23 ) / 4 \) (5b15 + 2b14 + 2b13 + b12 - 3b11 - b10 - b9 + b8 - 4b7 - 2b6 + 3b5 - 2b4 + 7b3 - 4b2 - 3*b1 - 23) / 4
\(\nu^{8}\)	\(=\)	\( ( \beta_{15} + 4 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + \cdots - 5 ) / 4 \) (b15 + 4b14 + 2b13 - 3b12 + 3b11 + 3b10 + 3b9 + 11b8 + 2b7 - 2b6 - 7b5 + 4b4 + 3b3 - 22b2 - 15b1 - 5) / 4
\(\nu^{9}\)	\(=\)	\( ( \beta_{15} + 10 \beta_{14} + 2 \beta_{13} + 13 \beta_{12} - 3 \beta_{11} - \beta_{10} + 7 \beta_{9} + \cdots - 15 ) / 4 \) (b15 + 10b14 + 2b13 + 13b12 - 3b11 - b10 + 7b9 - 3b8 + 14b6 - 13b5 + 2b4 + 3b3 + 8b2 + 5b1 - 15) / 4
\(\nu^{10}\)	\(=\)	\( ( 17 \beta_{15} + 8 \beta_{14} - 6 \beta_{13} - 19 \beta_{12} - 17 \beta_{11} - 13 \beta_{10} - 5 \beta_{9} + \cdots - 17 ) / 4 \) (17b15 + 8b14 - 6b13 - 19b12 - 17b11 - 13b10 - 5b9 - 17b8 + 14b7 + 14b6 - 11b5 + 16b4 + 19b3 - 2b2 + 17*b1 - 17) / 4
\(\nu^{11}\)	\(=\)	\( ( - 3 \beta_{15} + 22 \beta_{14} - 22 \beta_{13} - 7 \beta_{12} - 11 \beta_{11} - 5 \beta_{10} + \cdots + 33 ) / 4 \) (-3b15 + 22b14 - 22b13 - 7b12 - 11b11 - 5b10 + 3b9 + 17b8 - 28b7 - 26b6 - 25b5 - 6b4 - 9b3 - 8b2 + 9*b1 + 33) / 4
\(\nu^{12}\)	\(=\)	\( ( - 31 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 13 \beta_{10} + \cdots - 125 ) / 4 \) (-31b15 - 4b14 + 18b13 + 5b12 + 3b11 - 13b10 + 3b9 + 11b8 - 6b7 - 10b6 - 7b5 - 20b4 - 29b3 - 158b2 + 17*b1 - 125) / 4
\(\nu^{13}\)	\(=\)	\( ( - 11 \beta_{15} + 18 \beta_{14} - 38 \beta_{13} + 17 \beta_{12} + 41 \beta_{11} - 29 \beta_{10} + \cdots - 131 ) / 4 \) (-11b15 + 18b14 - 38b13 + 17b12 + 41b11 - 29b10 - 5b9 + 5b8 - 40b7 + 6b6 - 13b5 + 54b4 - 17b3 + 20b2 + 17*b1 - 131) / 4
\(\nu^{14}\)	\(=\)	\( ( - 15 \beta_{15} + 16 \beta_{14} + 2 \beta_{13} - 27 \beta_{12} + 15 \beta_{11} - 69 \beta_{10} + \cdots + 167 ) / 4 \) (-15b15 + 16b14 + 2b13 - 27b12 + 15b11 - 69b10 + 75b9 - 17b8 + 46b7 - 58b6 - 51b5 + 56b4 - 45b3 + 126b2 + 73*b1 + 167) / 4
\(\nu^{15}\)	\(=\)	\( ( - 51 \beta_{15} + 6 \beta_{14} - 118 \beta_{13} + 57 \beta_{12} - 67 \beta_{11} - 125 \beta_{10} + \cdots + 105 ) / 4 \) (-51b15 + 6b14 - 118b13 + 57b12 - 67b11 - 125b10 - 53b9 - 71b8 + 28b7 + 6b6 - 9b5 - 22b4 - 41b3 + 56b2 + 113*b1 + 105) / 4

\(n\)	\(1471\)	\(1765\)	\(3137\)	\(4609\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2353.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$5$	\( T^{16} + 48 T^{14} + \cdots + 64 \) T^16 + 48T^14 + 902T^12 + 8384T^10 + 40313T^8 + 96864T^6 + 101584T^4 + 29888*T^2 + 64
$7$	\( T^{16} \) T^16
$11$	\( T^{16} + 76 T^{14} + \cdots + 2304 \) T^16 + 76T^14 + 2190T^12 + 29900T^10 + 194425T^8 + 528720T^6 + 563808T^4 + 205056*T^2 + 2304
$13$	\( T^{16} + 108 T^{14} + \cdots + 1149184 \) T^16 + 108T^14 + 4182T^12 + 71692T^10 + 582033T^8 + 2221888T^6 + 4161888T^4 + 3646464*T^2 + 1149184
$17$	\( (T^{8} - 76 T^{6} + \cdots + 256)^{2} \) (T^8 - 76T^6 + 104T^5 + 1544T^4 - 3744T^3 - 1184T^2 + 896T + 256)^2
$19$	\( T^{16} + 140 T^{14} + \cdots + 11723776 \) T^16 + 140T^14 + 7478T^12 + 197300T^10 + 2784513T^8 + 21381384T^6 + 84436048T^4 + 136476160*T^2 + 11723776
$23$	\( (T^{8} + 4 T^{7} + \cdots + 25472)^{2} \) (T^8 + 4T^7 - 120T^6 - 568T^5 + 3544T^4 + 18176T^3 - 17792T^2 - 96384*T + 25472)^2
$29$	\( T^{16} + 192 T^{14} + \cdots + 589824 \) T^16 + 192T^14 + 14086T^12 + 492304T^10 + 8249353T^8 + 58264896T^6 + 145390080T^4 + 29048832*T^2 + 589824
$31$	\( (T^{8} - 12 T^{7} + \cdots - 145241)^{2} \) (T^8 - 12T^7 - 90T^6 + 1168T^5 + 2448T^4 - 32204T^3 - 24462T^2 + 222264*T - 145241)^2
$37$	\( T^{16} + \cdots + 13545235456 \) T^16 + 324T^14 + 39174T^12 + 2275324T^10 + 67664817T^8 + 995045944T^6 + 6339030480T^4 + 16189502976*T^2 + 13545235456
$41$	\( (T^{8} - 148 T^{6} + \cdots - 29696)^{2} \) (T^8 - 148T^6 + 352T^5 + 5936T^4 - 25344T^3 - 12992T^2 + 128512T - 29696)^2
$43$	\( T^{16} + \cdots + 326726560000 \) T^16 + 372T^14 + 54566T^12 + 4099172T^10 + 170784449T^8 + 3965288880T^6 + 48310589920T^4 + 258970229504*T^2 + 326726560000
$47$	\( (T^{8} - 12 T^{7} + \cdots - 2304)^{2} \) (T^8 - 12T^7 - 60T^6 + 1048T^5 - 568T^4 - 23328T^3 + 54816T^2 + 21888*T - 2304)^2
$53$	\( T^{16} + \cdots + 1973491776 \) T^16 + 304T^14 + 37174T^12 + 2358576T^10 + 82621705T^8 + 1541139312T^6 + 12750847632T^4 + 19458467136*T^2 + 1973491776
$59$	\( T^{16} + \cdots + 3515066944 \) T^16 + 476T^14 + 90910T^12 + 8856236T^10 + 459972409T^8 + 11969866592T^6 + 125121353104T^4 + 262062897728*T^2 + 3515066944
$61$	\( T^{16} + \cdots + 10909384704 \) T^16 + 408T^14 + 52048T^12 + 2937856T^10 + 79768576T^8 + 1023442944T^6 + 5719130112T^4 + 13627293696*T^2 + 10909384704
$67$	\( T^{16} + \cdots + 499498389504 \) T^16 + 636T^14 + 148534T^12 + 16011508T^10 + 830877313T^8 + 20407019544T^6 + 231892894224T^4 + 1017100924416*T^2 + 499498389504
$71$	\( (T^{8} - 20 T^{7} + \cdots + 8590080)^{2} \) (T^8 - 20T^7 - 60T^6 + 3912T^5 - 23512T^4 - 107424T^3 + 1681824T^2 - 6526848*T + 8590080)^2
$73$	\( (T^{8} - 4 T^{7} + \cdots - 1022064)^{2} \) (T^8 - 4T^7 - 290T^6 + 340T^5 + 27137T^4 + 33048T^3 - 756072T^2 - 2231136*T - 1022064)^2
$79$	\( (T^{8} - 4 T^{7} + \cdots - 37585)^{2} \) (T^8 - 4T^7 - 290T^6 + 2080T^5 + 8104T^4 - 63876T^3 - 5670T^2 + 140984*T - 37585)^2
$83$	\( T^{16} + \cdots + 29873665600 \) T^16 + 588T^14 + 114462T^12 + 10092844T^10 + 425922873T^8 + 7641426736T^6 + 33130946256T^4 + 53846383296*T^2 + 29873665600
$89$	\( (T^{8} - 336 T^{6} + \cdots + 1114112)^{2} \) (T^8 - 336T^6 - 576T^5 + 21120T^4 + 34816T^3 - 354304T^2 - 196608T + 1114112)^2
$97$	\( (T^{8} - 12 T^{7} + \cdots - 2509616)^{2} \) (T^8 - 12T^7 - 338T^6 + 2956T^5 + 31313T^4 - 237448T^3 - 729192T^2 + 5650592*T - 2509616)^2
show more
show less